FLC ON AN LTT TOOK EVASIVE ACTION IN RESPONSE TO TCASII RA AND THE SUBSEQUENT SIGHTING OF ANOTHER ACFT.

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ASSIGNED 3000 FT; APPROX 25 DME NW OF MIA VOR; WE WERE FLYING OUR SHORTS 360 DIRECT TO THE MIA VOR. WE RECEIVED A TCASII RESOLUTION ADVISORY TO DSND WHICH THE FO PROPERLY RESPONDED TO. IMMEDIATELY; WE RECEIVED A CONFLICTING COMMAND FROM THE TCASII 'CLB NOW; CLB NOW.' UPON TERMINATING THE DSCNT AND INITIATING THE CLB; I SAW THE NAV AND LNDG LIGHTS OF AN ACFT CLBING TOWARDS US IN A WESTERLY OR NWESTERLY HDG. WE DEVIATED 800 FT IN ALT CLBING FROM 3000 FT TO 3800 FT. WE IMMEDIATELY NOTIFIED APCH AND RETURNED TO 3000 FT. THIS OCCURRED JUST AFTER WE HAD BEEN HANDED OFF FROM ONE CTLR TO THE NEXT AND WE HAD NOT YET CHKED ON WITH THE NEW CTLR. UPON ARR IN MIAMI; I CALLED APCH CTL. THE SUPVR WAS WAITING FOR MY PHONE CALL AND EXPLAINED TO ME THAT THERE WAS A PIPER NAVAJO THAT HAD JUST DEPARTED OPA LOCKA CLBING VFR WBOUND OUT OF 2800 FT FOR 6500 FT WHEN WE HAD OUR CLOSE ENCOUNTER. WE WERE NEVER ADVISED OF ANY ACFT WITHIN OUR VICINITY. THE FO SAID OUR TCASII READ THAT NAVAJO TO BE 100 FT BELOW US. THE SUPVR I SPOKE WITH WAS EVIDENTLY PRESENT WHEN THIS OCCURRED BECAUSE HE TOLD ME THAT ACCORDING TO HIS EQUIP; THERE WAS ROUGHLY 300 FT OF VERT SEPARATION. MY CONCERN HERE IS TWOFOLD: ONE REGARDING TFC ADVISORIES AND THE OTHER THAT WE RECEIVED CONFLICTING TCASII COMMANDS.